Working in Radians with Exact Values

On Paper 1 (non-calculator), circular-measure answers are expected in terms of $\pi$. That means converting degrees to radians as a fraction of $\pi$, keeping it simplified, and never reaching for a decimal. The formulae are the same; the discipline is exactness.

A sector has radius $9$ cm and an angle of $120^\circ$ at the centre. (i) Express $120^\circ$ in radians, in terms of $\pi$. [1] (ii) Find the exact arc length. [2] (iii) Find the exact area of the sector. [2]

The working

(i) Convert with $\times \dfrac{\pi}{180}$: $120^\circ = 120 \times \frac{\pi}{180} = \frac{120\pi}{180} = \frac{2\pi}{3} \text{ radians} \quad \text{(B1)}$

(ii) Arc length $s = r\theta$ with $r = 9$, $\theta = \frac{2\pi}{3}$: $s = 9 \times \frac{2\pi}{3} = \frac{18\pi}{3} = 6\pi \text{ cm} \quad \text{(M1, A1)}$

(iii) Sector area $A = \frac12 r^2\theta$: $A = \frac{1}{2}(9)^2 \times \frac{2\pi}{3} = \frac{1}{2}(81)\times\frac{2\pi}{3} = \frac{81\pi}{3} = 27\pi \text{ cm}^2 \quad \text{(M1, A1)}$

Where the marks are won and lost

• Simplify the radian fraction. $\frac{120\pi}{180}$ must reduce to $\frac{2\pi}{3}$; an unsimplified angle usually forfeits the exact-value marks downstream.
• Keep $\pi$ symbolic throughout. $6\pi$ and $27\pi$ are the intended answers; $18.8$ and $84.8$ would lose accuracy marks on an “exact” question.
• Cancel cleanly: $9 \times \frac{2\pi}{3} = 6\pi$ (the $9$ and $3$ cancel), and $\frac{81 \times 2}{3\times 2} \cdot \pi$ tidies to $27\pi$.

Common mistakes

• Leaving the angle as $\frac{120\pi}{180}$ or converting to a decimal.
• Using degrees directly in $r\theta$ (giving nonsense like $9 \times 120$).
• Dropping the $\pi$ from the final answers.

Full method: Radians notes. Topic home: Circular Measure pillar.